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~~Let, $$s=\sum_{n=0}^{m} r^2$$~~
So, $$s=1+r+r^2+...+r^m$$
$$rs=r+r^2+...r^{m+1}$$
$$rs=s-1+r^{m+1}$$
$$s(r-1)=-(1-r^{m+1})$$
$$s=\frac{1-r^{m+1}}{1-r}$$
You can try the method to solve this kind of problem.

Let, $$s=\sum_{n=0}^{m} r^n$$
So, $$s=1+r+r^2+...+r^m$$
$$rs=r+r^2+...r^{m+1}$$
$$rs=s-1+r^{m+1}$$
$$s(r-1)=-(1-r^{m+1})$$
$$s=\frac{1-r^{m+1}}{1-r}$$
You can try the method to solve this kind of problem.